How to interpret lower order interaction when higher order interaction is significant? I have a question about the interpretation of lower order interaction terms in the presence of a significant higher order interaction effect.
Suppose I have a 2 (factor $A$) $\times$ 2 (factor $B$) $\times$ 2 (factor $C$) design where the highest order interaction ($A\times B\times C$) is significant and a lower order interaction term ($A\times B$) is also significant. Does the significant $A\times B\times C$ interaction render the $A\times B$ interaction uninterpretable (much like how main effects are rendered uninterpretable in the presence of a significant interaction)? 
Under this kind of circumstance, should I run a set of post-hoc/planned comparisons to check how different conditions are different?
 A: It's not that lower order interactions or main effects are completely uninterpretable when there's a higher order interaction.  It's that their interpretation is qualified.  For example, a main effect of A just means that overall, within the range of the IVs you have chosen, the levels of A differ; but that the magnitude, or possibly direction, of the difference really does vary across the other variables.  So, not completely uninterpretable, just qualified.  Upon a study of the data you may find you really do believe there is the main effect in general that's just varying in magnitude.  Or, you may find that the main effect only exists for a limited range of conditions.
In your case You have an AxB interaction and the magnitude of the interaction is dependent upon C.  It's possible the direction is as well but that's relatively unlikely.  The AxB interaction you have suggests how to examine the three way.  Make 2 2x2 AxB interaction plots, one at each level of C.  Your 3-way interaction tells you that the differences in these two interaction plots are worth noting.
A: You might want to run two 2-way, AxB ANOVAs, one for each level of the C variable. That approach will let you look at the "simple" two way interaction effects unaffected by the 3 way interaction.  Since there was a statistically significant 3 way interaction, we expect that the two simple 2 way interactions will not look the same.
